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Banach spaces whose algebras of operators have a large group of unitary elements

Published online by Cambridge University Press:  01 January 2008

JULIO BECERRA GUERRERO
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain. e-mail: [email protected]
MARÍA BURGOS
Affiliation:
Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Facultad de Ciencias Experimentales, 04120-Almería, Spain. e-mail: [email protected]; [email protected]
EL AMIN KAIDI
Affiliation:
Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Facultad de Ciencias Experimentales, 04120-Almería, Spain. e-mail: [email protected]; [email protected]
ÁNGEL RODRÍGUEZ PALACIOS
Affiliation:
Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Facultad de Ciencias Experimentales, 04120-Almería, Spain. e-mail: [email protected]; [email protected]

Abstract

We prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution • on satisfying T = T−1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach space X is a Hilbert space if and only if it is a real JB*-triple and is -unitary, where stands for the dual weak-operator topology.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2008

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